As part of our attack on the protein-folding problem, we have been studying the mechanism of secondary-structure formation in proteins. Our current approach is to analyze the thermodynamics and kinetics of the formation of individual structural elements in model peptides in the context of simple statistical mechanical models. We have studied the kinetics of beta-hairpin formation in a 16-residue peptide within a simple physical picture in which structures are classified according to backbone conformation as defined by 15 pairs of dihedral angles. Each dihedral-angle pair is assumed to have only two possible conformational states, the native (beta-hairpin) state and the nonnative (random- coil)state. The formation of the beta-hairpin structure of the peptide is opposed by the entropy loss associated with fixing successive dihedral-angle pairs in the native conformation, and favored by native side chain interactions that arise in sufficiently long contiguous stretches of native dihedral-angle pairs. We have implemented a complete model description of the equilibrium and kinetic properties of the peptide, requiring 32768 states. However, we have shown that a much simpler single-sequence approximation, in which only the 121 states having at most a single contiguous stretch of native dihedral-angle pairs are considered, is adequate to describe the kinetics of beta- hairpin formation in the peptide measured using temperature-jump techniques. We have also used a similar model approach to treat the kinetics of alpha-helix formation in model peptides of various lengths. In this case, helical structures in the peptide consist of sufficiently long contiguous stretches of dihedral-angle pairs in a native (helical) conformation; the native conformation of individual dihedral-angle pairs is stabilized by native hydrogen bonds with other residues in the same helical stretch, and destabilized by entropy losses. Here again the single-sequence approximation, which considers only 254of the 4 million possible states of a 21-residue peptide, predicts all the major features of the measured kinetics, while at the same time fitting both the circular dichroism and fluorescence equilibrium melting curves. We have also undertaken theoretical/computational studies of the kinetics of secondary-structure formation, including the development of efficient algorithms for simulating at the atomic level the folding dynamics of peptides in solution. - protein folding, kinetic models, secondary structure formation, circular dichroism, fluorescence, temperature-jump